D-branes in non-abelian gauged linear sigma models

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1 D-branes in non-abelian gauged linear sigma models Johanna Knapp Vienna University of Technology Bonn, September 29, 2014

2 Outline CYs and GLSMs D-branes in GLSMs Conclusions

3 CYs and GLSMs A Calabi-Yau space (CY) can be realized as the low energy configuration of a supersymmetric N = (2, 2) gauge theory in two dimensions - the gauged linear sigma model (GLSM). [Witten 93] The choice of gauge groups and matter content determines the CY. Hypersurfaces and complete intersections in toric ambient spaces are realized by GLSMs with gauge groups G = U(1) k. Non-abelian gauge groups lead to exotic CYs (e.g. Pfaffian CYs). Phase: by tuning coupling constants (FI parameters) in the gauge theory, we can probe the (enlarged) Kähler moduli space of a CY.

4 GLSM Data Gauge group G = U(1)l H {±1,±1}. This talk: H = USp(k) or H = O(k), l = 1. Matter Chiral fields p i, i = 1,..., M, U(1) l -charges Q l p, trivial rep. of H i xi a, i = 1,..., N, a = 1,..., k, U(1) l -charges Qx l i, fundamental rep. of H Twisted chiral fields σ k, take values in the maximal torus of G. Superpotential W H = SU(2) : W = H = O(2) : W = N A ij (p)[x i x j ] = i,j=1 N S ij (p)(x i x j ) = i,j=1 N 2 i,j=1 a,b=1 N 2 i,j=1 a,b=1 A ij (p)x a i ε ab x b j S ij (p)x a i δ ab x b j

5 Equations of motion I The vacua, i.e. solutions to the equations of motion, determine the CY. On the Higgs branch σ = 0. D-terms U(1) l : M N Q l p i pi 2 + Qx l j x j 2 = r l i=1 j=1 SU(2) : xx 1 2 x = 0 O(2) : xx (xx ) T = 0. r l R are the FI parameters. Together with the θ-angle they can be identified with the complexified Kähler modulus of the CY: t = r iθ. By changing the value of the r l we can probe the Kähler moduli space of the CY.

6 Equations of motion II E.o.ms continued... F-terms W p i = 0 W x a i = 0. Phases of the GLSM Different FI parameter regions have different solutions of the e.o.ms. Gauge group gets broken to a subgroup in different phases. strong coupling effects if broken to continuous subgroup CYs in phases are not necessarily birational, yet share the same Kähler moduli space. Conjecture: The derived categories associated to the CYs in the various phases are equivalent.

7 CY data CY condition Dimension: M Q l p + k N Q l i x j = 0 rkh = k i=1 dim = M 1 j=1 k(k ± 1) USp(k),... O(k) Hodge number estimate h 1,1 number of FI-parameters h 2,1 number of monomials in W modulo reparametrization Quantum corrections on the Coulomb branch determine the conifold point(s).

8 Non-abelian Duality There is a duality between models with different non-abelian gauge groups. [Hori 11] original dual G G = U(1) l USp(k) G = U(1) l USp(N k 1) fields W p i x i p i x i a ij = a ji Q l Q l p i x i Q l Q l p i x i Q xi + Q xj W = N i,j=1 Aij (p)[x i x j ] W = N i,j=1 [ x i x j ]a ij + A ij (p)a ij Similar structure for orthogonal gauge groups with O(k) SO(N k + 1) s ij = s ji Strong/weak coupling duality: Certain properties of the CY are more easily derived in the dual theory.

9 Non-abelian examples Notation: Models A k q and S k, q A k q: H = USp(k), with charges q S k, q : H = SO(k) ( = 0) or H = O(k) SO(k) Z 2 ( = ±) with charges q Rodland Model A 2 2 7,1 7 [Rodland 98][Hori-Tong 06] r 0: Complete intersection of codim. 7 in G(2, 7) r 0: Pfaffian CY rka(p) = 4 (strongly coupled) Hosono-Takagi S 2,+ 2 5,1 5 [Hosono-Takagi][Hori 11] r 0: Complete intersection of codim.5 in (P 4 P 4 )/Z 2 r 0: Determinantal variety rks(p) 4 (strongly coupled) Further interesting new one-parameter examples. [Hori-JK, 13] CYs with different Hodge numbers h 2,1 but same Kähler moduli space.

10 Matrix Factorizations D-branes are gauge-invariant, R-invariant matrix factorizations of the GLSM potential. [Herbst-Hori-Page 08][Honda-Okuda,Hori-Romo 13] Take a square matrix Q with polynomial entries such that Gauge invariance: R-invariance: Q 2 = W 1 ρ(g) 1 Q(gφ)ρ(g) = Q(φ) λ r Q(λ R φ)λ r = λq(φ) GLSM branes map to (trivial or non-trivial) branes in the individual phases.

11 Elementary matrix factorizations I To construct matrix factorizations for the Rodland and HT-examples we start with a toy model with gauge group U(2). Superpotential W = p[x 1 x 2 ] = p(x1 1 x2 2 x1 2 x2 1 ). ( ) a b Gauge transformations with g = U(2): c d p (ad bc) 1 p xi 1 axi 1 + bxi 2 xi 2 cxi 1 + dxi 2. R-charge: can choose [p] = 2, [x a i ] = 0.

12 Elementary matrix factorizations II Simplest one: ( Q = 0 p x 1 1 x 2 2 x 2 1 x ) ρ = ( det g ) Using a basis of 4 4 Clifford matrices {η i, η j } = δ ij : Q = x 1 1 η 1 + px 2 2 η 1 + x 2 1 η 2 px 1 2 η 2 Q = 0 0 x1 1 x px2 1 px2 2 px2 2 x px2 1 x ρ = det g d 0 0 det g 0 0 b det g c det g a det g For the models of interest we can obtain matrix factorizations by taking tensor products of these two elementary constructions.

13 D-brane charges Quantum corrected D-brane central charges are computed by the hemisphere partition function. [Honda-Okuda,Hori-Romo 13] Z D 2(B) = C d l G τ α(τ) sinh(iπα(τ)) ( Γ Q i (τ) + R ) i e t R (τ) f B(τ) γ 2 α>0 i α > 0 positive roots τ = t C with t C = Lie(T G ) C (maximal torus) l G = rk(g) R i... R-charges, Q i... weights under the maximal torus t R = r iθ... complexified FI (Kähler) parameter γ... integration contour (s.t. integral is convergent) Brane factor ( f B (τ) = tr M ( 1) r e 2πiρ (τ)) M... Chan-Paton space The brane input is obtained by restricting the matrices ρ(g) and λ r to the maximal torus.

14 Structure Sheaf In geometric phases Z D 2 reduces to Z D 2(B) = ˆΓ(X )e B+ i 2π ω ch(b). X The structure sheaf encodes the topological data of the CY. Z(O X ) = mirror = H 3 3! ( ) 3 ( ) it it c2 H + 2π 2π 24 + i ζ(3) (2π) 3 χ(x ) + O(e t ) H 3 3! ϖ 3 + c 2H 24 ϖ 2 + i ζ(3)χ(x ) (2π) 3 ϖ 0 Different topological data for different (geometric) phases in the GLSM. Identify the brane factor/matrix factorization associated to the structure sheaf.

15 Structure Sheaf II For Rodland and HT matrix factorizations discussed above lead to a brane factors of type f B (a, b) = (1 e 2πi(τ1+τ2) ) a (1 e 2πiτ1 ) b (1 e 2πiτ2 ) b In r 0-phases the structure sheaf is given by the matrix factorization with brane factor f (N, 0) Q(N, 0) = N i=1 p i η i + W p i η i a, b Z 0 In the strongly-coupled r 0-phases no simple matrix factorization has been found. However, There are brane factors that give the right charges, e.g. for Rodland: f B = 2f (5, 1) 2f (3, 2) + f (6, 1) 2f (4, 2) f (5, 2) In the weakly-coupled dual theory the structure sheaf is described by a simple matrix factorization.

16 Conclusions We constructed matrix factorizations in examples of one-parameter non-abelian GLSMs. Using the hemisphere partition function we computed quantum-corrected D-brane charges. We identified the structure sheaf. We found many examples of lower-dimensional branes in all phases. These methods are very general. Not restricted to hypersurfaces/complete intersections in toric spaces. No mirror symmetry required. No Landau-Ginzburg point required for matrix factorizations.

17 Outlook D-brane transport By choosing integration contours such that Z D 2 converges one can derive a grade restriction rule. [HHP 08] This reproduces mathematical results. [Addington-Donovan-Segal 14] For strongly coupled phases, branes have to be grade restricted deep inside the phase. How do construct matrix factorizations for the structure sheaf in strongly coupled phases? How to classify matrix factorizations in (non-)abelian GLSMs? Interesting phenomenology on these CYs?